## An Efficient Spectral Method for Ordinary Differential Equations with Rational Function Coefficients |

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Airy equation algorithms appropriately restricted domain approximate solution approximation to multiplication assume banded representation bandwidth boundary value problems bounded direction bounded inverse Chebyshev expansion Chebyshev polynomials classical orthogonal polynomials coefficient matrix computation condition number constraints convergence Coutsias David Torres denote DEPARTMENT OF MATHEMATICS EFFICIENT SPECTRAL eigenfunctions eigenvalue error estimates example exist constants expansion coefficients Extreme singular values finite Fourier mode Galerkin approximation Gegenbauer polynomials Hermite polynomials homogeneous problem integration operator Jacobi polynomials Laplace operator linear differential operators lower bound LU decomposition nonsingular nonzero elements norms null space obtain operators with rational order of truncation ordinary differential equations orthogonal family orthogonal polynomial families parameter pentadiagonal polynomials of degree Proof properties Qo~n r-method rational coefficients rational function coefficients rational maps recurrence relation rows satisfy singular Sturm-Liouville solve spectral methods Sturm-Liouville problems subspace Theorem 2.1 Thomas Hagstrom vector zero